Fisher transformation

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A graph of the transformation. The x axis is before transformation, the y axis after. There is also the identity transformation for comparison. The effects are small until abs(x) > 0.3.
A graph of the transformation. The x axis is before transformation, the y axis after. There is also the identity transformation for comparison. The effects are small until abs(x) > 0.3.

In statistics, hypotheses about the value of the population correlation coefficient ρ between variables X and Y of the underlying population, can be tested using the Fisher transformation applied to the sample correlation r.

[edit] Definition of the Fisher transformation

Let N be the sample size. The transformation is defined by:

z = {1 \over 2}\log{1+r \over 1-r} = \operatorname{arctanh}(r).

If (X, Y) has a bivariate normal distribution, then z is approximately normally distributed with mean

{1 \over 2}\log{{1+\rho} \over {1-\rho}},

and standard deviation

{1 \over \sqrt{N-3}}.

This transformation, and its inverse,

{{\exp(2z)-1} \over {\exp(2z)+1}},

are a common way of constructing a confidence interval for ρ.

[edit] Discussion

The behavior of this transform has been extensively studied since Fisher introduced it in 1915. Fisher himself found the exact distribution of Zr for data from a bivariate normal distribution in 1921; Gayen, 1951 determined the exact distribution of Zr for data from a bivariate Type A Edgeworth distribution. Hotelling in 1953 calculated the Taylor series expressions for the moments of Zr and several related statistics and Hawkins in 1989 discovered the asymptotic distribution of Zr for virtually any data.

[edit] References

  • Fisher, R.A. (1915). Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population. Biometrika, 10, 507-521.
  • Fisher, R.A. (1921). On the `probable error' of a coefficient of correlation deduced from a small sample. Metron, 1, 3-32.
  • Hawkins, D.L. (1989). Using U statistics to derive the asymptotic distribution of Fisher's Z statistic. The American Statistician, 43, 235-237.